\(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x)^4 (c i+d i x)^2} \, dx\) [201]

Optimal result

Integrand size = 45, antiderivative size = 729 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4 (c i+d i x)^2} \, dx=-\frac {2 A B d^4 n (a+b x)}{(b c-a d)^5 g^4 i^2 (c+d x)}+\frac {2 B^2 d^4 n^2 (a+b x)}{(b c-a d)^5 g^4 i^2 (c+d x)}-\frac {12 b^2 B^2 d^2 n^2 (c+d x)}{(b c-a d)^5 g^4 i^2 (a+b x)}+\frac {b^3 B^2 d n^2 (c+d x)^2}{(b c-a d)^5 g^4 i^2 (a+b x)^2}-\frac {2 b^4 B^2 n^2 (c+d x)^3}{27 (b c-a d)^5 g^4 i^2 (a+b x)^3}-\frac {2 B^2 d^4 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d)^5 g^4 i^2 (c+d x)}-\frac {12 b^2 B d^2 n (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 g^4 i^2 (a+b x)}+\frac {2 b^3 B d n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 g^4 i^2 (a+b x)^2}-\frac {2 b^4 B n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 (b c-a d)^5 g^4 i^2 (a+b x)^3}+\frac {d^4 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^5 g^4 i^2 (c+d x)}-\frac {6 b^2 d^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^5 g^4 i^2 (a+b x)}+\frac {2 b^3 d (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^5 g^4 i^2 (a+b x)^2}-\frac {b^4 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 (b c-a d)^5 g^4 i^2 (a+b x)^3}-\frac {4 b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{3 B (b c-a d)^5 g^4 i^2 n} \] Output:

-2*A*B*d^4*n*(b*x+a)/(-a*d+b*c)^5/g^4/i^2/(d*x+c)+2*B^2*d^4*n^2*(b*x+a)/(- 
a*d+b*c)^5/g^4/i^2/(d*x+c)-12*b^2*B^2*d^2*n^2*(d*x+c)/(-a*d+b*c)^5/g^4/i^2 
/(b*x+a)+b^3*B^2*d*n^2*(d*x+c)^2/(-a*d+b*c)^5/g^4/i^2/(b*x+a)^2-2/27*b^4*B 
^2*n^2*(d*x+c)^3/(-a*d+b*c)^5/g^4/i^2/(b*x+a)^3-2*B^2*d^4*n*(b*x+a)*ln(e*( 
(b*x+a)/(d*x+c))^n)/(-a*d+b*c)^5/g^4/i^2/(d*x+c)-12*b^2*B*d^2*n*(d*x+c)*(A 
+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^5/g^4/i^2/(b*x+a)+2*b^3*B*d*n*(d* 
x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^5/g^4/i^2/(b*x+a)^2-2/9* 
b^4*B*n*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^5/g^4/i^2/(b* 
x+a)^3+d^4*(b*x+a)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^5/g^4/i^2/ 
(d*x+c)-6*b^2*d^2*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^5/g 
^4/i^2/(b*x+a)+2*b^3*d*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b 
*c)^5/g^4/i^2/(b*x+a)^2-1/3*b^4*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^ 
2/(-a*d+b*c)^5/g^4/i^2/(b*x+a)^3-4/3*b*d^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n)) 
^3/B/(-a*d+b*c)^5/g^4/i^2/n
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1695\) vs. \(2(729)=1458\).

Time = 2.57 (sec) , antiderivative size = 1695, normalized size of antiderivative = 2.33 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4 (c i+d i x)^2} \, dx =\text {Too large to display} \] Input:

Integrate[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/((a*g + b*g*x)^4*(c*i + 
 d*i*x)^2),x]
 

Output:

-1/27*(36*b*B^2*d^3*n^2*(a + b*x)^3*(c + d*x)*Log[(a + b*x)/(c + d*x)]^3 + 
 9*B*n*Log[(a + b*x)/(c + d*x)]^2*(12*a^3*A*b*c*d^3 + b^4*B*c^4*n - 6*a*b^ 
3*B*c^3*d*n + 18*a^2*b^2*B*c^2*d^2*n - 3*a^4*B*d^4*n + 36*a^2*A*b^2*c*d^3* 
x + 12*a^3*A*b*d^4*x - 2*b^4*B*c^3*d*n*x + 18*a*b^3*B*c^2*d^2*n*x + 36*a^2 
*b^2*B*c*d^3*n*x - 12*a^3*b*B*d^4*n*x + 36*a*A*b^3*c*d^3*x^2 + 36*a^2*A*b^ 
2*d^4*x^2 + 6*b^4*B*c^2*d^2*n*x^2 + 54*a*b^3*B*c*d^3*n*x^2 + 12*A*b^4*c*d^ 
3*x^3 + 36*a*A*b^3*d^4*x^3 + 22*b^4*B*c*d^3*n*x^3 + 18*a*b^3*B*d^4*n*x^3 + 
 12*A*b^4*d^4*x^4 + 10*b^4*B*d^4*n*x^4 + 12*b*B*d^3*(a + b*x)^3*(c + d*x)* 
Log[e*((a + b*x)/(c + d*x))^n] - 12*b*B*d^3*n*(a + b*x)^3*(c + d*x)*Log[(a 
 + b*x)/(c + d*x)]) + 3*b*d^2*(b*c - a*d)*(a + b*x)^2*(c + d*x)*(27*A^2 + 
78*A*B*n + 92*B^2*n^2 + 27*B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 - 6*B*n*(9 
*A + 13*B*n)*Log[(a + b*x)/(c + d*x)] + 27*B^2*n^2*Log[(a + b*x)/(c + d*x) 
]^2 + 6*B*Log[e*((a + b*x)/(c + d*x))^n]*(9*A + 13*B*n - 9*B*n*Log[(a + b* 
x)/(c + d*x)])) + 6*b*d^3*(a + b*x)^3*(c + d*x)*Log[a + b*x]*(18*A^2 + 30* 
A*B*n + 55*B^2*n^2 + 18*B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 - 6*B*n*(6*A 
+ 5*B*n)*Log[(a + b*x)/(c + d*x)] + 18*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 
+ 6*B*Log[e*((a + b*x)/(c + d*x))^n]*(6*A + 5*B*n - 6*B*n*Log[(a + b*x)/(c 
 + d*x)])) + b*(b*c - a*d)^3*(c + d*x)*(9*A^2 + 6*A*B*n + 2*B^2*n^2 + 9*B^ 
2*Log[e*((a + b*x)/(c + d*x))^n]^2 - 6*B*n*(3*A + B*n)*Log[(a + b*x)/(c + 
d*x)] + 9*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 + 6*B*Log[e*((a + b*x)/(c ...
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 522, normalized size of antiderivative = 0.72, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2961, 2795, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{(a g+b g x)^4 (c i+d i x)^2} \, dx\)

\(\Big \downarrow \) 2961

\(\displaystyle \frac {\int \frac {(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^4}d\frac {a+b x}{c+d x}}{g^4 i^2 (b c-a d)^5}\)

\(\Big \downarrow \) 2795

\(\displaystyle \frac {\int \left (\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 d^4-\frac {4 b (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 d^3}{a+b x}+\frac {6 b^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 d^2}{(a+b x)^2}-\frac {4 b^3 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 d}{(a+b x)^3}+\frac {b^4 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^4}\right )d\frac {a+b x}{c+d x}}{g^4 i^2 (b c-a d)^5}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {b^4 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 (a+b x)^3}-\frac {2 b^4 B n (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{9 (a+b x)^3}+\frac {2 b^3 d (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{(a+b x)^2}+\frac {2 b^3 B d n (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{(a+b x)^2}-\frac {6 b^2 d^2 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{a+b x}-\frac {12 b^2 B d^2 n (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}+\frac {d^4 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{c+d x}-\frac {2 A B d^4 n (a+b x)}{c+d x}-\frac {4 b d^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{3 B n}-\frac {2 b^4 B^2 n^2 (c+d x)^3}{27 (a+b x)^3}+\frac {b^3 B^2 d n^2 (c+d x)^2}{(a+b x)^2}-\frac {12 b^2 B^2 d^2 n^2 (c+d x)}{a+b x}-\frac {2 B^2 d^4 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x}+\frac {2 B^2 d^4 n^2 (a+b x)}{c+d x}}{g^4 i^2 (b c-a d)^5}\)

Input:

Int[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/((a*g + b*g*x)^4*(c*i + d*i*x 
)^2),x]
 

Output:

((-2*A*B*d^4*n*(a + b*x))/(c + d*x) + (2*B^2*d^4*n^2*(a + b*x))/(c + d*x) 
- (12*b^2*B^2*d^2*n^2*(c + d*x))/(a + b*x) + (b^3*B^2*d*n^2*(c + d*x)^2)/( 
a + b*x)^2 - (2*b^4*B^2*n^2*(c + d*x)^3)/(27*(a + b*x)^3) - (2*B^2*d^4*n*( 
a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/(c + d*x) - (12*b^2*B*d^2*n*(c + 
d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x) + (2*b^3*B*d*n*(c + 
 d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x)^2 - (2*b^4*B*n*( 
c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*(a + b*x)^3) + (d^4* 
(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(c + d*x) - (6*b^2*d^2 
*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(a + b*x) + (2*b^3*d* 
(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(a + b*x)^2 - (b^4*( 
c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*(a + b*x)^3) - (4* 
b*d^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/(3*B*n))/((b*c - a*d)^5*g^ 
4*i^2)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2795

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + 
(e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[ 
c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b 
, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0 
] && IntegerQ[m] && IntegerQ[r]))
 

rule 2961

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol 
] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q   Subst[Int[x^m*((A + B*L 
og[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[ 
b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3055\) vs. \(2(721)=1442\).

Time = 48.60 (sec) , antiderivative size = 3056, normalized size of antiderivative = 4.19

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(3056\)

Input:

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^4/(d*i*x+c*i)^2,x,method 
=_RETURNVERBOSE)
 

Output:

1/27*(36*B^2*x^4*ln(e*((b*x+a)/(d*x+c))^n)^3*b^9*d^7+108*A^2*x^4*ln(e*((b* 
x+a)/(d*x+c))^n)*b^9*d^7+180*A*B*x^3*b^9*c*d^6*n^2+108*B^2*x^2*ln(e*((b*x+ 
a)/(d*x+c))^n)^3*a*b^8*c*d^6+54*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)^2*b^9*c^ 
2*d^5*n+648*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^7*d^7*n^2+198*B^2*x^2* 
ln(e*((b*x+a)/(d*x+c))^n)*b^9*c^2*d^5*n^2+480*B^2*x^2*a*b^8*c*d^6*n^3+324* 
A*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)^2*a^2*b^7*d^7-342*A*B*x^2*a^2*b^7*d^7*n^ 
2+198*A*B*x^2*b^9*c^2*d^5*n^2+108*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^3*a^2*b^ 
7*c*d^6-108*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a^3*b^6*d^7*n-18*B^2*x*ln(e* 
((b*x+a)/(d*x+c))^n)^2*b^9*c^3*d^4*n+216*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)*a 
^3*b^6*d^7*n^2-30*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)*b^9*c^3*d^4*n^2-87*B^2*x 
*a^2*b^7*c*d^6*n^3+567*B^2*x*a*b^8*c^2*d^5*n^3+324*A^2*x^2*ln(e*((b*x+a)/( 
d*x+c))^n)*a*b^8*c*d^6+216*A^2*x^2*a*b^8*c*d^6*n+108*A*B*x*ln(e*((b*x+a)/( 
d*x+c))^n)^2*a^3*b^6*d^7-114*A*B*x*a^3*b^6*d^7*n^2-30*A*B*x*b^9*c^3*d^4*n^ 
2-54*B^2*a^4*b^5*d^7*n^3+2*B^2*b^9*c^4*d^3*n^3-27*A^2*a^4*b^5*d^7*n+9*A^2* 
b^9*c^4*d^3*n+972*A*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^8*c*d^6*n+648*A*B* 
x*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^7*c*d^6*n-330*A*B*a^3*b^6*c*d^6*n^2+324* 
A*B*a^2*b^7*c^2*d^5*n^2-54*A*B*a*b^8*c^3*d^4*n^2+162*B^2*x*ln(e*((b*x+a)/( 
d*x+c))^n)^2*a*b^8*c^2*d^5*n+648*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^7*c 
*d^6*n^2+486*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)*a*b^8*c^2*d^5*n^2+324*A*B*x*l 
n(e*((b*x+a)/(d*x+c))^n)^2*a^2*b^7*c*d^6-216*A*B*x*ln(e*((b*x+a)/(d*x+c...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3183 vs. \(2 (721) = 1442\).

Time = 0.17 (sec) , antiderivative size = 3183, normalized size of antiderivative = 4.37 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4 (c i+d i x)^2} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^4/(d*i*x+c*i)^2,x 
, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4 (c i+d i x)^2} \, dx=\text {Timed out} \] Input:

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))**2/(b*g*x+a*g)**4/(d*i*x+c*i)** 
2,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6171 vs. \(2 (721) = 1442\).

Time = 0.49 (sec) , antiderivative size = 6171, normalized size of antiderivative = 8.47 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4 (c i+d i x)^2} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^4/(d*i*x+c*i)^2,x 
, algorithm="maxima")
 

Output:

-1/3*B^2*((12*b^3*d^3*x^3 + b^3*c^3 - 5*a*b^2*c^2*d + 13*a^2*b*c*d^2 + 3*a 
^3*d^3 + 6*(b^3*c*d^2 + 5*a*b^2*d^3)*x^2 - 2*(b^3*c^2*d - 8*a*b^2*c*d^2 - 
11*a^2*b*d^3)*x)/((b^7*c^4*d - 4*a*b^6*c^3*d^2 + 6*a^2*b^5*c^2*d^3 - 4*a^3 
*b^4*c*d^4 + a^4*b^3*d^5)*g^4*i^2*x^4 + (b^7*c^5 - a*b^6*c^4*d - 6*a^2*b^5 
*c^3*d^2 + 14*a^3*b^4*c^2*d^3 - 11*a^4*b^3*c*d^4 + 3*a^5*b^2*d^5)*g^4*i^2* 
x^3 + 3*(a*b^6*c^5 - 3*a^2*b^5*c^4*d + 2*a^3*b^4*c^3*d^2 + 2*a^4*b^3*c^2*d 
^3 - 3*a^5*b^2*c*d^4 + a^6*b*d^5)*g^4*i^2*x^2 + (3*a^2*b^5*c^5 - 11*a^3*b^ 
4*c^4*d + 14*a^4*b^3*c^3*d^2 - 6*a^5*b^2*c^2*d^3 - a^6*b*c*d^4 + a^7*d^5)* 
g^4*i^2*x + (a^3*b^4*c^5 - 4*a^4*b^3*c^4*d + 6*a^5*b^2*c^3*d^2 - 4*a^6*b*c 
^2*d^3 + a^7*c*d^4)*g^4*i^2) + 12*b*d^3*log(b*x + a)/((b^5*c^5 - 5*a*b^4*c 
^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)* 
g^4*i^2) - 12*b*d^3*log(d*x + c)/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^ 
3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*g^4*i^2))*log(e*(b*x 
/(d*x + c) + a/(d*x + c))^n)^2 - 2/3*A*B*((12*b^3*d^3*x^3 + b^3*c^3 - 5*a* 
b^2*c^2*d + 13*a^2*b*c*d^2 + 3*a^3*d^3 + 6*(b^3*c*d^2 + 5*a*b^2*d^3)*x^2 - 
 2*(b^3*c^2*d - 8*a*b^2*c*d^2 - 11*a^2*b*d^3)*x)/((b^7*c^4*d - 4*a*b^6*c^3 
*d^2 + 6*a^2*b^5*c^2*d^3 - 4*a^3*b^4*c*d^4 + a^4*b^3*d^5)*g^4*i^2*x^4 + (b 
^7*c^5 - a*b^6*c^4*d - 6*a^2*b^5*c^3*d^2 + 14*a^3*b^4*c^2*d^3 - 11*a^4*b^3 
*c*d^4 + 3*a^5*b^2*d^5)*g^4*i^2*x^3 + 3*(a*b^6*c^5 - 3*a^2*b^5*c^4*d + 2*a 
^3*b^4*c^3*d^2 + 2*a^4*b^3*c^2*d^3 - 3*a^5*b^2*c*d^4 + a^6*b*d^5)*g^4*i...
 

Giac [F(-1)]

Timed out. \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4 (c i+d i x)^2} \, dx=\text {Timed out} \] Input:

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^4/(d*i*x+c*i)^2,x 
, algorithm="giac")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 32.83 (sec) , antiderivative size = 3157, normalized size of antiderivative = 4.33 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4 (c i+d i x)^2} \, dx=\text {Too large to display} \] Input:

int((A + B*log(e*((a + b*x)/(c + d*x))^n))^2/((a*g + b*g*x)^4*(c*i + d*i*x 
)^2),x)
 

Output:

log(e*((a + b*x)/(c + d*x))^n)*((x*(8*A*B*b^2*c*d - 8*A*B*a*b*d^2 + 12*B^2 
*b*d*n*(a*d + b*c) + (16*B^2*a*b*d^2*n)/3 - (16*B^2*b^2*c*d*n)/3) - 6*A*B* 
a^2*d^2 + 2*A*B*b^2*c^2 + 6*B^2*a^2*d^2*n + (2*B^2*b^2*c^2*n)/3 + 12*B^2*b 
^2*d^2*n*x^2 + 4*A*B*a*b*c*d + (16*B^2*a*b*c*d*n)/3)/(x*(3*a^6*d^4*g^4*i^2 
 - 9*a^2*b^4*c^4*g^4*i^2 + 24*a^3*b^3*c^3*d*g^4*i^2 - 18*a^4*b^2*c^2*d^2*g 
^4*i^2) - x^2*(9*a*b^5*c^4*g^4*i^2 - 9*a^5*b*d^4*g^4*i^2 - 18*a^2*b^4*c^3* 
d*g^4*i^2 + 18*a^4*b^2*c*d^3*g^4*i^2) - x^3*(3*b^6*c^4*g^4*i^2 - 9*a^4*b^2 
*d^4*g^4*i^2 + 24*a^3*b^3*c*d^3*g^4*i^2 - 18*a^2*b^4*c^2*d^2*g^4*i^2) + x^ 
4*(3*a^3*b^3*d^4*g^4*i^2 - 3*b^6*c^3*d*g^4*i^2 + 9*a*b^5*c^2*d^2*g^4*i^2 - 
 9*a^2*b^4*c*d^3*g^4*i^2) - 3*a^3*b^3*c^4*g^4*i^2 + 3*a^6*c*d^3*g^4*i^2 + 
9*a^4*b^2*c^3*d*g^4*i^2 - 9*a^5*b*c^2*d^2*g^4*i^2) - (4*d^3*(6*A*B*b + 5*B 
^2*b*n)*(x*((a*d + b*c)*((3*a*g^4*i^2*n*(a*d - b*c)^4)/(2*d) + (3*g^4*i^2* 
n*(a*d - b*c)^4*(2*a*d - b*c))/(2*d^2)) + (3*a*b*c*g^4*i^2*n*(a*d - b*c)^4 
)/d) + x^2*(b*d*((3*a*g^4*i^2*n*(a*d - b*c)^4)/(2*d) + (3*g^4*i^2*n*(a*d - 
 b*c)^4*(2*a*d - b*c))/(2*d^2)) + (3*b*g^4*i^2*n*(a*d + b*c)*(a*d - b*c)^4 
)/d) + a*c*((3*a*g^4*i^2*n*(a*d - b*c)^4)/(2*d) + (3*g^4*i^2*n*(a*d - b*c) 
^4*(2*a*d - b*c))/(2*d^2)) + 3*b^2*g^4*i^2*n*x^3*(a*d - b*c)^4))/(3*g^4*i^ 
2*n*(a*d - b*c)^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(x*(3*a^6*d^4*g^4*i^2 - 
9*a^2*b^4*c^4*g^4*i^2 + 24*a^3*b^3*c^3*d*g^4*i^2 - 18*a^4*b^2*c^2*d^2*g^4* 
i^2) - x^2*(9*a*b^5*c^4*g^4*i^2 - 9*a^5*b*d^4*g^4*i^2 - 18*a^2*b^4*c^3*...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 5699, normalized size of antiderivative = 7.82 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4 (c i+d i x)^2} \, dx =\text {Too large to display} \] Input:

int((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^4/(d*i*x+c*i)^2,x)
 

Output:

( - 324*log(a + b*x)*a**6*b*c*d**4*n - 324*log(a + b*x)*a**6*b*d**5*n*x - 
108*log(a + b*x)*a**5*b**2*c**2*d**3*n - 324*log(a + b*x)*a**5*b**2*c*d**4 
*n**2 - 1080*log(a + b*x)*a**5*b**2*c*d**4*n*x - 324*log(a + b*x)*a**5*b** 
2*d**5*n**2*x - 972*log(a + b*x)*a**5*b**2*d**5*n*x**2 - 396*log(a + b*x)* 
a**4*b**3*c**2*d**3*n**2 - 324*log(a + b*x)*a**4*b**3*c**2*d**3*n*x - 810* 
log(a + b*x)*a**4*b**3*c*d**4*n**3 - 1368*log(a + b*x)*a**4*b**3*c*d**4*n* 
*2*x - 1296*log(a + b*x)*a**4*b**3*c*d**4*n*x**2 - 810*log(a + b*x)*a**4*b 
**3*d**5*n**3*x - 972*log(a + b*x)*a**4*b**3*d**5*n**2*x**2 - 972*log(a + 
b*x)*a**4*b**3*d**5*n*x**3 - 510*log(a + b*x)*a**3*b**4*c**2*d**3*n**3 - 1 
188*log(a + b*x)*a**3*b**4*c**2*d**3*n**2*x - 324*log(a + b*x)*a**3*b**4*c 
**2*d**3*n*x**2 - 2940*log(a + b*x)*a**3*b**4*c*d**4*n**3*x - 2160*log(a + 
 b*x)*a**3*b**4*c*d**4*n**2*x**2 - 648*log(a + b*x)*a**3*b**4*c*d**4*n*x** 
3 - 2430*log(a + b*x)*a**3*b**4*d**5*n**3*x**2 - 972*log(a + b*x)*a**3*b** 
4*d**5*n**2*x**3 - 324*log(a + b*x)*a**3*b**4*d**5*n*x**4 - 1530*log(a + b 
*x)*a**2*b**5*c**2*d**3*n**3*x - 1188*log(a + b*x)*a**2*b**5*c**2*d**3*n** 
2*x**2 - 108*log(a + b*x)*a**2*b**5*c**2*d**3*n*x**3 - 3960*log(a + b*x)*a 
**2*b**5*c*d**4*n**3*x**2 - 1512*log(a + b*x)*a**2*b**5*c*d**4*n**2*x**3 - 
 108*log(a + b*x)*a**2*b**5*c*d**4*n*x**4 - 2430*log(a + b*x)*a**2*b**5*d* 
*5*n**3*x**3 - 324*log(a + b*x)*a**2*b**5*d**5*n**2*x**4 - 1530*log(a + b* 
x)*a*b**6*c**2*d**3*n**3*x**2 - 396*log(a + b*x)*a*b**6*c**2*d**3*n**2*...